In this section it shall be assumed that the position and orientation of the wheel in space is known. For this one measures the position and velocity of the wheel from the vehicle chassis, see Fig.7.19. The position of the wheel carrier can be described through the wheel center point R and through the introduced wheel carrier fixed coordinate system introduced in Chap.6. These values are known through the vehicle kinematics. What is now required is the position and velocity of the wheel-road contact zone, which in this case is given by a contact point A and an orientation of a tangential plane on the road surface in point A.
Figure7.20shows a schematic representation, in which the wheel center plane is represented as a flat disc. The road surface is described locally through the tangential planeR. This is for the description of a normal road surface without steps or other individual obstacles completely sufficient.
Given are a road surface normal vectorns and the wheel normal vectornR as well as the position of the wheel center point R, given by the position vector in the inertial system. The position of the wheel is, in this case, as described in the previous chapter, described through the corresponding geometrical and kinemat- ical relationships originating from the vehicle chassis over the wheel suspension to the wheel carrier. With these values one can now determine the position of the wheel contact point A. This point is given as the contact point between a rigid disc (center point R, radius as yet unknown) and the road surfaceR. The connecting vector from R to A must be orthogonal to both the wheel axisnRas well as the unit vectornLin the longitudinal direction of the wheel disc (Fig.7.21).
It is advantageous for the process of understanding, to create a mechanical substitute model for the vectors at the wheel (Fig.7.22). It consists of a wheel carrier R mounted rotating on the angle element W1, whose lower arm is per- pendicular to the rotation axis. This lower arm thus always lies in the wheel plane.
Using a translational joint it is connected to a second, angle element W2 lying vertically on the floor. This is again orthogonal, such that the connecting line from R and A as required not only lies in the wheel plane, but also is perpendicular to the longitudinal direction ofnL. At the same time, the plate located on the floor is always pointed in the direction ofnL. The average tire radiusR, i.e. the length of
the distance AR, relates to the displacement of the translational joint. For a given position and orientation of the wheel carrier R, the arms of the angle point in such a way, that between the link of W2and the road is a line contact. The plate located on the road surface, which represents the contact patch, can rotate about this link relative to the angle W2. The presented kinematic structure can also be interpreted as a kinematic chain with six joint DoF, which represents the spatial pose of the wheel carrier relative to the road surface. The calculation of the vectors of this system is now done successively:
Spanning out a tripodfR;nR;nL;nRAgin the wheel surfaceThe unit vectors in the rolling direction of the wheel is given by
nLẳ nRnS
knRnSk: ð7:51ị The unit vector from R to A is given by:
nRAẳnRnL: ð7:52ị For the unit vector at right angles to the rolling direction one arrives at (Fig.7.21):
vehicle chassis
tire-road contact point Fig. 7.19 Calculation of the position and velocity of the wheel carrier from the vehicle chassis
(camber)
Fig. 7.20 Unit vector at the wheel
rotational joint
rotational joint
translational joint wheel carrier
plate
Fig. 7.21 Multibody replacement model of the wheel geometry (Schnelle1990)
nQẳnSnL: ð7:53ị Derivation of the current wheel radiusRThe position vector to the point A is given in the inertial system. Its projection on the normal to the road surface nR
gives the height of the road surface (in more detail it gives the height of the point A and hence that of the tangential planeR) in the inertial system. This is known and has the valueu. Hence one can derive:
RẳubnS nRAnS
: ð7:54ị
With the position vectorrRof the wheel center point R, the position vectorrAto the point A is given by:
wheel carrier
Fig. 7.22 Velocities of the wheel road contact
rAẳrRỵRnRA: ð7:55ị Furthermore the position of the two angle elements W1 and W2 is known through the unit vectorsnRA;nLandnQ. There now exists two coordinate systems, one wheel carrier fixed through R;f nL;nR;nRAg as well as a patch fixed system given by A; nL;nQ;nS
. In this way one can define the camber anglecbetween the wheel plane and the vertical plane. The following simplified relationships are enough to describe the relative rotation of the coordinate systems
coscẳnRnQ; sincẳ nRnS: ð7:56ị Iterative evaluation of the contact point for uneven road surfacesIf the road surface is uneven, one can no longer assume a constant road surface normal vector nS. This is more over dependent on the position, i.e. the position vector rA. If however the road surfaces are constrained to be lightly curved, the contact point is clearly definable as described above and also the directions can be specified with respect to the tangential plane with a good approximation in the following manner:
• As a starting value forrA, the projection of the wheel center R on the surface (in the direction of gravity) is chosen.
• The road surface normal vector nS is formed at this point.
• With this vectornSand according to (7.51)–(7.55) the contact point geometry is constructed. Even the road height u is part of the iteration, as it is also dependent on the position.
• The valuerA calculated from (7.55) is now a better estimation. The iteration is started again with the new value ofrA. The iteration is continued so long till the difference between two successive values ofrAlies within a pre-defined iteration tolerance. At this point it is not appropriate to set too high standards in the precision, as the errors made in this step are to be viewed in relation with the errors already induced due to the simplification during the modeling of the tires.